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The Cost of Failure: On The Complexity of Recampaigning under Fixed Districts

Michael C. Chavrimootoo, Aidan Jeansonne

TL;DR

The paper studies recampaigning: strategically assigning new candidates to fixed districts to maximize wins when redistricting is not possible. It formalizes multiple variants with and without budgets, and with bounded or unbounded numbers of winners, then analyzes their complexity under common voting rules. A central result shows that 1-Recampaigning is solvable in polynomial time for rules with the $\,1$-Winner condition via a reduction to Min-Cost Unbalanced Assignment, while for $\\ell>1$ and natural scoring families, many cases are NP-hard, though some trivial or resolute rules remain tractable. The work also establishes fixed-parameter tractability for bounded recampaigning with respect to the number of districts and discusses hardness and separations among priced/unpriced and bounded/unbounded variants, outlining rich directions for future exploration such as destructive recampaigning and tighter complexity gaps.

Abstract

Redistricting efforts have gathered contemporary attention in both quotidian and scholarly debates, particularly in the United States where efforts to redraw congressional districts to favor either of the two major parties in 12 states -- such as California, Texas, and Ohio -- have captured the public eye. The treatment of redistricting in computational social choice has essentially focused on the process of determining "appropriate" districts. In this work, we are interested in understanding the gamut of options left for the "losing" party, and so we consider the flip side of the problem: Given fixed/predetermined districts, can a given party still make their candidates win by strategically placing them in certain districts? We dub this as "recampaigning" to capture the intuition that a party would redirect their campaigning efforts from one district to another. We model recampaigning as a computational problem, consider natural variations of the model, and study those new models through the lens of (1) (polynomial-time many-one) interreducibilities, (2) separations/collapses (both unconditional and axiomatic-sufficient), and (3) both worst-case and parametrized complexity.

The Cost of Failure: On The Complexity of Recampaigning under Fixed Districts

TL;DR

The paper studies recampaigning: strategically assigning new candidates to fixed districts to maximize wins when redistricting is not possible. It formalizes multiple variants with and without budgets, and with bounded or unbounded numbers of winners, then analyzes their complexity under common voting rules. A central result shows that 1-Recampaigning is solvable in polynomial time for rules with the -Winner condition via a reduction to Min-Cost Unbalanced Assignment, while for and natural scoring families, many cases are NP-hard, though some trivial or resolute rules remain tractable. The work also establishes fixed-parameter tractability for bounded recampaigning with respect to the number of districts and discusses hardness and separations among priced/unpriced and bounded/unbounded variants, outlining rich directions for future exploration such as destructive recampaigning and tighter complexity gaps.

Abstract

Redistricting efforts have gathered contemporary attention in both quotidian and scholarly debates, particularly in the United States where efforts to redraw congressional districts to favor either of the two major parties in 12 states -- such as California, Texas, and Ohio -- have captured the public eye. The treatment of redistricting in computational social choice has essentially focused on the process of determining "appropriate" districts. In this work, we are interested in understanding the gamut of options left for the "losing" party, and so we consider the flip side of the problem: Given fixed/predetermined districts, can a given party still make their candidates win by strategically placing them in certain districts? We dub this as "recampaigning" to capture the intuition that a party would redirect their campaigning efforts from one district to another. We model recampaigning as a computational problem, consider natural variations of the model, and study those new models through the lens of (1) (polynomial-time many-one) interreducibilities, (2) separations/collapses (both unconditional and axiomatic-sufficient), and (3) both worst-case and parametrized complexity.
Paper Structure (13 sections, 22 theorems, 9 equations)

This paper contains 13 sections, 22 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Elections and Voting Rules
  4. Our Models
  5. Relationships and Properties
  6. Worst-Case Complexity
  7. 1-Recampaigning
  8. Unbounded and $\ell$-Recampaigning, for $\ell > 1$
  9. Parametrizing by Number of Districts
  10. Conclusion
  11. Modeling.
  12. Relationships.
  13. Complexity.

Key Result

Proposition 2.1

For each voting rule $\mathcal{E}$, $W_\mathcal{E} \leq_m^p \mathcal{E}\hbox{-}{\rm CRC}_{\mathbb{N}}$.

Theorems & Definitions (47)

  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Definition 3.2: $k$-Resolute
  • Proposition 3.3
  • proof
  • Proposition 3.5
  • proof
  • Definition 3.6
  • ...and 37 more